This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 84

2014 ASDAN Math Tournament, 10
Tags: 2014 , Geometry Test
In a convex quadrilateral $ABCD$ we are given that $\angle CAD=10^\circ$, $\angle DBC=20^\circ$, $\angle BAD=40^\circ$, $\angle ABC=50^\circ$. Compute angle $BDC$.

2014 ASDAN Math Tournament, 5
Tags: 2014 , Geometry Test
Consider a triangle $ABC$ with $AB=4$, $BC=3$, and $AC=2$. Let $D$ be the midpoint of line $BC$. Find the length of $AD$.

2014 ASDAN Math Tournament, 7
Tags: 2014 , General Test
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.

2014 ASDAN Math Tournament, 7
Tags: 2014 , team test
Eddy draws $6$ cards from a standard $52$-card deck. What is the probability that four of the cards that he draws have the same value?

2014 ASDAN Math Tournament, 8
Tags: 2014 , General Test
George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi, and vegetable jiaozi. Each person orders exactly $15$ jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiaozi in both orders.

2014 ASDAN Math Tournament, 10
Tags: 2014 , team test
Three real numbers $x$, $y$, and $z$ are chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that $x$, $y$, and $z$ can be the side lengths of a triangle.

2014 ASDAN Math Tournament, 8
Tags: 2014 , Geometry Test
Moor made a lopsided ice cream cone. It turned out to be an oblique circular cone with the vertex directly above the perimeter of the base (see diagram below). The height and base radius are both of length $1$. Compute the radius of the largest spherical scoop of ice cream that it can hold such that at least $50\%$ of the scoop’s volume lies inside the cone. [center]<see attached>[/center]

2014 ASDAN Math Tournament, 23
Tags: 2014 , General Test
Let triangle $ABC$ have side lengths $AB=11$, $BC=7$, and $AC=12$. Let $D$ be a point on $AC$ and $E$ be a point on $AB$ such that $\angle CDE=90^\circ$ and the area of triangle $CDE$ is maximized. Find the area of triangle $CDE$.

1989 Bundeswettbewerb Mathematik, 4
Tags: induction , strong induction , combinatorics proposed , combinatorics , Taiwan TST , 2014
Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

2014 ASDAN Math Tournament, 2
Tags: 2014 , Geometry Test
Let $ABC$ be a triangle with sides $AB=19$, $BC=21$, and $AC=20$. Let $\omega$ be the incircle of $ABC$ with center $I$. Extend $BI$ so that it intersects $AC$ at $E$. If $\omega$ is tangent to $AC$ at the point $D$, then compute the length of $DE$.

2014 ASDAN Math Tournament, 3
Tags: 2014 , Geometry Tiebreaker
Let $ABC$ be a triangle and $I$ its incenter. Suppose $AI=\sqrt{2}$, $BI=\sqrt{5}$, $CI=\sqrt{10}$ and the inradius is $1$. Let $A'$ be the reflection of $I$ across $BC$, $B'$ the reflection across $AC$, and $C'$ the reflection across $AB$. Compute the area of triangle $A'B'C'$.

2014 ASDAN Math Tournament, 9
Tags: 2014 , General Test
The operation $\oslash$, called "reciprocal sum," is useful in many areas of physics. If we say that $x=a\oslash b$, this means that $x$ is the solution to $$\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$$ Compute $4\oslash2\oslash4\oslash3\oslash4\oslash4\oslash2\oslash3\oslash2\oslash4\oslash4\oslash3$.

2014 ASDAN Math Tournament, 5
Tags: 2014 , General Test
Screws are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$?

2014 ASDAN Math Tournament, 22
Tags: 2014 , General Test
Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.

2014 ASDAN Math Tournament, 11
Tags: 2014 , team test
In the following system of equations $$|x+y|+|y|=|x-1|+|y-1|=2,$$ find the sum of all possible $x$.

2014 ASDAN Math Tournament, 21
Tags: 2014 , General Test
A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s?

2014 ASDAN Math Tournament, 13
Tags: 2014 , General Test
Square $S_1$ is inscribed inside circle $C_1$, which is inscribed inside square $S_2$, which is inscribed inside circle $C_2$, which is inscribed inside square $S_3$, which is inscribed inside circle $C_3$, which is inscribed inside square $S_4$. [center]<see attached>[/center] Let $a$ be the side length of $S_4$, and let $b$ be the side length of $S_1$. What is $\tfrac{a}{b}$?

2014 ASDAN Math Tournament, 3
Tags: 2014 , team test
A segment of length $1$ is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region.

2014 ASDAN Math Tournament, 3
Tags: 2014 , Algebra Test
Compute all prime numbers $p$ such that $8p+1$ is a perfect square.

2014 ASDAN Math Tournament, 10
Tags: 2014 , Algebra Test
Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.

2014 ASDAN Math Tournament, 5
Tags: 2014 , Algebra Test
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.

2014 ASDAN Math Tournament, 19
Tags: 2014 , General Test
Given that $f(x)+2f(4-x)=x+8$, compute $f(16)$.

2014 ASDAN Math Tournament, 7
Tags: 2014 , Algebra Test
$f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$.

2014 Contests, 1
Tags: trigonometry , induction , algebra , Vietnam , 2014
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and \[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$. Prove that they are converges and find their limits.

2014 ASDAN Math Tournament, 1
Tags: 2014 , Advanced Topics Tiebreaker
Compute the number of three digit numbers such that all three digits are distinct and in descending order, and one of the digits is a $5$.